Apparatus and method for producing an annular far-field patterned beam(s)

ABSTRACT

A phased array of Gaussian beams was used to form an annular far-field pattern as opposed to the normal peaked pattern in most phased arrays. This annular pattern arises from the radial symmetry of the polarization in each beam. Beams opposite one another in the annulus are polarized in the same direction but are 180° out of phase. This causes the center portion of the beam to be a null rather than a peak.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

The invention(s) described herein may be manufactured and used by or forthe government of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefore.

FIELD OF THE INVENTION

This invention relates to the formation of a composite laser beam(s)that exhibit(s) a far field intensity pattern having a central null, asopposed to a central peak.

BACKGROUND

Coherently phased arrays have been in use in radar systems for manyyears. In these radar systems the phase of two or more antennas arelocked together and the ensemble forms a diffraction pattern that can bealtered by changing the antenna spacing, antenna amplitude, and relativephase relationships Analogous optical systems have recently been formedby actively or passively locking the phases of two or more identicaloptical beams. By adjusting the relative phases of each beam, thediffraction pattern formed by the optical ensemble can be changed.

The ability to adjust the intensity pattern of light in the far fieldsimply by changing the relative phase or intensity has tremendousutility. Two useful consequences of this phase-intensity relationshipare far-field beam steering and synthetic aperture shaping. Beamsteering using phase control gives the possibility of fast,non-mechanical beam deflection. Synthetic aperture devices can be usedto dynamically control focal length, multiple beams from the sameaperture, and aberration correction all of which arise from the abilityto arbitrarily control far-field intensity patterns using spatiallyregistered phase control.

Phase control of coherent arrays is the most common method of far-fieldintensity shaping, but polarization control also plays a role in theintensity distribution. Standard coherently combined arrays employ beamswhich are all polarized along the same axis. In this case thepolarization is simply a necessary condition for achieving a phasedarray. However if one allows the polarization to become a parameter,other possibilities arise.

One of the possibilities of polarization control in addition to phasecontrol is the ability to form what are referred to here as discretecylindrical vector (DCV) beams. Cylindrical vector (CV) beams havecharacteristics that distinguish them from Gaussian beams. Among thesecharacteristics is a nonuniform polarization state across the beam.Namely, the polarization of CV beams vary uniformly around the center ofthe beam and can be radially polarized, azimuthally polarized, or somecombination of the two. Additionally, CV beams have a characteristicnull in the center of the beam which is required by the cylindricalsymmetry. Applications of CV beams range from mitigation of thermaleffects in high power lasers and laser machining to particleacceleration interactions. They can even be used to generatelongitudinal electric fields when tightly focused due to the cylindricalsymmetry of the polarization, and although they are typically formed infree space laser cavities using some variation of an axicon, they canalso be formed and guided in fibers.

Typical methods for creating CV beams fall into two categories. Thefirst category of techniques uses an intracavity axicon in a laserresonator to generate a CV mode. The second category starts with asingle beam and rotates the polarization of portions of the beam tocreate an inhomogeneously (typically radially or azimuthally) polarizedbeam. This second method is used by Biss et al. in U.S. Pat. No.7,151,632. The second method is also used by Schuster in U.S. Pat. No.6,392,800 to create a minimally perturbative transformation from asingle uniformly polarized beam to a radially polarized beam formicrolithographic projection.

A method performed in accordance with the principles of the presentinvention introduces a third possibility for creating CV beams. The saidmethod uses multiple beams to produce a composite beam whichapproximates a CV beam. A great utility offered by this novel method isthat it can be used to generate discrete CV beams synthetically by,among other things, superimposing an array of distinct beams havingsuccessively varying polarization, as is described in detail infra. Thepattern of beams is an approximation to a CV beam. Some of theadvantages of a synthetically created CV beam are similar to theadvantages of synthetic aperture radar over earlier radar techniques.For example, a synthetically generated CV beam created using a method inaccordance with the principles of the present invention (a DCV beam)exhibits many of the features as any other type of phased array, such asdynamically controlled beam steering, aberration correction, etc.

SUMMARY

The present invention relates to a method of producing a discretecylindrical vector beam. The method involves providing multiple beams.The multiple beams are manipulated to produce a least three manipulatedbeams. The manipulated beams are distributed in the near field such thatthe composite near field intensity distribution of said manipulatedbeams has a central null. Each manipulated beam is provided the sameshape as each other manipulated beam. Each manipulated beam is providedwith the same wavelength as each other manipulated beam. The manipulatedbeams are oriented in the near field such that each of said manipulatedbeams is uniformly distributed on the perimeter of a circle having aselected radius. Each of the manipulated beams is provided apolarization that differs from the polarization of its adjacentmanipulated beams by an amount equal to 360 degrees divided by a totalnumber of manipulated beams. The manipulated beams are propagated suchthat each manipulated beam is parallel to each other manipulated beam.The propagated manipulated beams produce a discrete cylindrical vectorbeam having a composite far field intensity distribution having acentral null

It is to be understood that the foregoing general description and thefollowing detailed description are explanatory only and are not to beviewed as being restrictive of the present invention, as claimed. Otherfeatures, embodiments, and aspects of a method performed in accordancewith the principles of the present invention will become apparent aftera review of the following detailed description of the disclosedembodiments, theoretical background, the drawings, and claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1. Figure one is a near field cross sectional view of one possibleembodiment of five manipulated beams produced in accordance with theprinciples of the present invention, located on the perimeter of acircle.

FIG. 2 is an illustration of one possible embodiment of an apparatusthat could be used to perform a method in accordance with the principlesof the present invention; however for ease of illustration, themanipulated beams are shown in a plane rather than oriented uniformly onthe perimeter of a circle as is required by the principles of thepresent invention.

FIG. 3 illustrates a near field intensity distribution of each of apossible embodiment of three manipulated beams, and a near fieldcomposite intensity distribution of one embodiment of three manipulatedbeams (including a central null), generated in accordance with theprinciples of the present invention.

FIG. 4 illustrates a near field intensity distribution of each of apossible embodiment of four manipulated beams, and a near fieldcomposite intensity distribution of one embodiment of four manipulatedbeams (including a central null), generated in accordance with theprinciples of the present invention.

FIG. 5 illustrates a near field intensity distribution of each of apossible embodiment of six manipulated beams, and a near field compositeintensity distribution of one embodiment of six manipulated beams(including a central null), generated in accordance with the principlesof the present invention.

FIG. 6 shows a simulation of a possible composite intensity distributionof a discrete cylindrical vector beam formed using one embodiment ofthree manipulated beams in accordance with the principles of the presentinvention.

FIG. 7 shows a simulation of a possible composite intensity distributionof a discrete cylindrical vector beam formed using one embodiment offour manipulated beams in accordance with the principles of the presentinvention.

FIG. 8 shows a simulation of a possible composite intensity distributionof a discrete cylindrical vector beam formed using one embodiment of sixmanipulated beams in accordance with the principles of the presentinvention.

FIG. 9 is an experimentally derived image of a far field compositeintensity distribution of a discrete cylindrical vector beam formedusing one possible embodiment of four manipulated beams in accordancewith the principles of the present invention, the discrete cylindricalvector beam having a central null in the far field.

FIG. 10 is an illustration of the preferred embodiment of a layout basedon fiber optic manipulation that may be used to perform a methodaccording to the principles of the present invention.

FIG. 11 is an experimentally derived image of a far field compositeintensity distribution of a discrete cylindrical vector beam formedusing one possible embodiment of six manipulated beams in accordancewith the principles of the present invention, the discrete cylindricalvector beam having a central null in the far field.

FIG. 12 is an experimentally derived image of a far field compositeintensity distribution of a discrete cylindrical vector beam formedusing one possible embodiment of three manipulated beams in accordancewith the principles of the present invention, the discrete cylindricalvector beam having a central null in the far field.

FIG. 13 is an illustration of how vectors may be rearranged and added toproduce the cancellation which is required for the central null.

FIG. 14 shows a simulation of the composite intensity distributionformed when three beams are propagated to produce far field effects whenthe near field beams all have the same polarization rather than thepolarization arrangement required by the principles of the presentinvention.

FIG. 15 is an experimentally derived image of a far field compositeintensity distribution formed using standard methods of coherentlycombining three near field beams with all having the same polarizationrather than the polarization arrangement required by the principles ofthe present invention, the composite beam does not have a central null.

FIG. 16 shows a simulation of the composite intensity distributionformed when four beams are propagated to produce far field effects whenthe near field beams all have the same polarization rather than thepolarization arrangement required by the principles of the presentinvention.

FIG. 17 is an experimentally derived image of a far field compositeintensity distribution formed using standard methods of coherentlycombining four near field beams with all having the same polarizationrather than the polarization arrangement required by the principles ofthe present invention, the composite beam does not have a central null.

FIG. 18 shows a simulation of the composite intensity distributionformed when six beams are propagated to produce far field effects whenthe near field beams all have the same polarization rather than thepolarization arrangement required by the principles of the presentinvention.

FIG. 19 is an experimentally derived image of a far field compositeintensity distribution formed using standard methods of coherentlycombining six near field beams with all having the same polarizationrather than the polarization arrangement required by the principles ofthe present invention, the composite beam does not have a central null.

DETAILED DESCRIPTION

One embodiment of a method practiced in accordance with the principlesof the present invention involves providing at least three multiplebeams (4 multiple beams are represented by lines in FIG. 2,8(a-d)—however, any number of at least three multiple beams can be usedin accordance with the principles of the present invention) andmanipulating the multiple beams to produce at least three manipulatedbeams FIG. 1, 2 (a-e); FIG. 2, 10 (a-d) (Note that FIG. 1 and FIG. 2show five and four manipulated beams respectively; however, any quantityof three or more manipulated beams may be used in accordance with theprinciples of the present invention). FIG. 1 illustrates a cross sectionof manipulated beams in the near field 2 (a-e). The shape, number, type,source, and other aspects of the multiple beams and manipulated beamsFIG. 1, 2 (a-e) may vary from those illustrated in accordance with theprinciples of the present invention, unless otherwise specified in thisspecification or required by principles or science or practice.

Each of the manipulated beams FIG. 1, 2 (a-e) is provided with the samephase as each of the other manipulated beams, meaning that φ_(n)=φ_(m)for any pair of manipulated beams n and m.

Each of the manipulated beams FIG. 1, 2 (a-e) is provided with the samewavelength as each of the other manipulated beams FIG. 1, 2 (a-e).

Each of the manipulated beams FIG. 1, 2 (a-e) is provided a polarizationê_(n) that varies from the polarization of each other manipulated beamFIG. 1, 2 (a-e). Specifically, the polarization ê_(n) of eachmanipulated beam varies FIG. 1, 2 (a-e) by 360 degrees divided by atotal number of manipulated beams FIG. 1, 2 (a-e) from that manipulatedbeam's adjacent manipulated beams. For instance, in FIG. 1, Beam 1's (2a) polarization would vary by 72 degrees (equivalent to 360 degreesdivided by 5—the total number of manipulated beams in the illustratedembodiment of manipulated beams in the near field) from the polarizationof each of Beams 2 (2 b) and 5 (2 e). Each manipulated beam has twoadjacent manipulated beams. For the purposes of this specification,including the claims, a first manipulated beam is adjacent to a secondmanipulated beam if the distance between the first manipulated beam andthe second manipulated beam is no greater than the distance between thesecond manipulated beam and any other manipulated beam when the distanceis measured along the perimeter of the circle (4) on which themanipulated beams 2 (a-e) lie in accordance with the principles of thepresent invention. For the purposes of this specification, including theclaims, each manipulated beam has exactly two adjacent manipulated beamsalong the perimeter of the said circle.

Each of the manipulated beams FIG. 1, 2 (a-e) is provided the samegeometrical shape as each of the other manipulated beams. FIG. 1illustrates one possible orientation of manipulated beams 2 (a-e) havinga non cylindrically symmetrical geometry. However, the manipulated beams2 (a-e) may have any geometrical shape, including cylindricallysymmetrical shapes.

The manipulated beams FIG. 1, 2 (a-e) are uniformly distributed on theperimeter of a single circle having a selected radius. Manipulated beamsare “uniformly distributed” on the perimeter of a circle (separated fromeach other by 360 degrees divided by the number of manipulated beams),for the purposes of this specification including the claims, when, foreach manipulated beam, the distance between a manipulated beam and itstwo adjacent multiple beams is the same where the distance betweenmanipulated beams is measured around the perimeter of the circle.

The manipulated beams are oriented on the perimeter of said circle suchthat the perimeter of the circle passes through an identical portion ofeach of said manipulated beams. FIG. 1 illustrates a cross section ofone possible embodiment of five manipulated beams 2 (a-e) distributedevenly on the perimeter of a single circle 4 having a radius d/2 (6). Ascan be seen from FIG. 1, the perimeter of the circle passes through anidentical portion of each of said manipulated beams. In FIG. 1, themanipulated beams 2 (a-e) do not have a cylindrically symmetricalgeometry, are oriented on the perimeter of a single circle 4 with aradius d/2 (6), and are uniformly distributed on the circle's perimeter4. However, a method performed in accordance with the principles of thepresent invention may include manipulated beams FIG. 1, 2 (a-e) havingany geometry, including hexagonal, rectangular, and square, as well asan arbitrary shape. Further, a method performed in accordance with theprinciples of the present invention includes providing distribution andorientation for the manipulated beams FIG. 1, 2 (a-e) such that they areuniformly distributed and oriented on the perimeter of a circle with anyselected radius (See FIG. 1) (so long as the rest of the requirementsare met—for example, the radius cannot be so small that the near fieldcomposite intensity distribution of the manipulated beams does not havea central null).

The manipulated beams FIG. 1, 2 (a-e) are further distributed such thattheir composite near field intensity distribution has a central null.The radius of said circle may affect whether the composite intensitydistribution has a central null. FIGS. 3, 4, and 5 illustrate thecomposite near field intensity distribution (18, 20, 22) of one possibleembodiment of three, four, and six manipulated beams respectively. Thenear field composite intensity distribution (18, 20, 22) of the fourmanipulated beams in the near field has a central null FIG. 5, 15. Thecentral null in FIG. 5 (15) consists of an area.

Each manipulated beam FIG. 1, 2 (a-e) is made to be parallel to eachother manipulated beam. Multiple beams (multiple beams are representedby lines in FIG. 2, 8 (a-d)) manipulated to yield manipulated beams FIG.1, 2 (a-e); FIG. 2, 10 (a-d), or manipulated beams, can be made parallelto each other by any conventional means. In one embodiment of a methodpracticed in accordance with the principles of the present invention,the multiple beams FIG. 2, 8 (a-d) are made parallel using a segmentedmirror (SM) 24. In reality (or practice) a perfectly parallel set ofmanipulated beams leading to a composite DCV beam cannot be created dueto the finite precision of any laboratory equipment, but in practicebeams can be made sufficiently parallel to create a functional DCV beam.As such, the use of the term parallel in this specification, includingthe claims, does not require perfect parallelism; the quality of the DCVbeam will depend on the accuracy with which each individual manipulatedbeam is made parallel to the other manipulated beams. Similar statementsmay be made about each of the quantities and requirements involved increating a DCV beam (for example, phasing, wavelength, intensity,distribution and orientation are all subject to limitations ofequipment).

The manipulated beams (which at this point have the same shape, samewavelength, same overall intensity, are in phase with each other, havebeen provided with differing polarizations as described and intensitydistributions, are parallel to each other, and lie on the perimeter of acircle with said orientation on said circle) are propagated to yield acomposite beam. The composite beam has a far field composite intensitydistribution. The far field composite intensity distribution has acentral null. A composite beam is the sum of the light from the array ofmanipulated beams FIG. 1, 2 (a-e), and it is considered to be a singleentity. A composite beam created using the principles of the presentinvention approximates a cylindrical vector beam in the far field. Asinitially separate beams are used to produce a composite beamapproximating a cylindrical vector beam, the cylindrical vector beamapproximated in accordance with the principles of the present inventionis referred to as a discrete cylindrical vector beam (DCV beam) in thisspecification including the claims.

FIGS. 3, 4, and 5 illustrate the intensity distribution of one possibleembodiment of three, four, and six manipulated beams in the near field(FIG. 3, 12 (a-c); FIG. 4, 14 (a-d); FIG. 5 16 (a-f), respectively), aswell as the near field composite intensity distribution of oneembodiment of three, four, and six manipulated beams (FIG. 3, 18; FIG.4, 20; FIG. 5, 22, respectively).

The far field composite intensity distributions of some possiblecomposite beams produced using some possible embodiments using three,four, and six manipulated beams are shown in FIGS. 6, 7, and 8. The onaxis fluence goes to zero at all points along the z-axis, producing acentral null (See FIG. 6, 26; FIG. 7, 28; FIG. 8, 30). On the z-axis,the distance from each aperture to any one point on the axis is alwaysthe same, and as mentioned previously, the phase is the same for eachmanipulated beam FIG. 1, 2 (a-e); FIG. 2, 10 (a-d).

FIG. 2 illustratively shows one possible apparatus constructed topractice one embodiment of a method in accordance with the principles ofthe present invention; the apparatus is used to provide and manipulatemultiple beams 8 (a-d) to produce manipulated beams FIG. 2, 10 (a-d).For illustrative purposes, FIG. 2 does not illustrate manipulated beamsthat are oriented around the perimeter of a circle as is required by theprinciples of the present invention.

In this particular embodiment, four manipulated beams are produced(represented by the lines in FIG. 2, 10 (a-d)); however, any number ofmanipulated beams greater than or equal to three may be used inaccordance with the principles of the present invention.

In this particular embodiment, the beam from the continuous wave Nd:YAGdoubled to 532 nm FIG. 2, 32 was expanded by means of a telescope (TS)to a diameter of about 1 cm. Any conventional beam source may be used.Further, any conventional beam diameter may be used. This beam isdirected through a diffractive optical element (DOE) 36 which creates an8×8 array of beams. Use of a diffractive optical element to providemultiple beams 8 (a-d) from a single beam is merely provided as anexample. Any method of providing multiple beams 8 (a-d) can be used inaccordance with the principles of this invention.

In this particular embodiment, each of the multiple beams that exits theDOE 36 has the same characteristics—geometry, wavelength, phase,intensity, polarization—as the input beam except that the propagationvector of each of the multiple beams 8 (a-d) is slightly different. Thisgives rise to a diverging array of identical multiple beams 8 (a-d).Further, in this particular embodiment, four beams are selected from thebeams exiting the diffractive optical element (from the 8*8 array)oriented such that they form a diverging 2 beam by 2 beam array. Thosebeams, which are propagating in a diverging square, are reflected andmade parallel by a segmented mirror (SM) 24, thereby providing fouridentical and parallel beams distributed and oriented uniformly aroundthe perimeter of a single circle. Any conventional method of makingparallel the multiple beams 8 (a-d) or otherwise providing parallelmanipulated beams may be used.

It is worth noting that in this embodiment, the intensity distribution(or shape of the beam) in the near field is cylindrically symmetrical.As such, the manipulated beams in this embodiment, unlike themanipulated beams shown in FIG. 1, 2 (a-e), did not need to be in effectrotated with respect to the z-axis in order to have the perimeter of thecircle pass through an identical portion of the manipulated beams.

In this particular embodiment, after the multiple beams have been madeparallel they pass through a set of four ½ wave plates (HWP) 38 whichare used to provide the described varying polarization for eachmanipulated beam. The manipulated beams then pass through articulatedglass slides (AGS) 40 which are used to provide each of said beams withthe same phase as each other of said beams, thereby yielding manipulatedbeams 10 (a-d).

For experimental purposes, the manipulated beams 10 (a-d) were focusedby a transform lens 42 in order to simulate the far field compositeintensity distribution at the lens focus. The lens 42 was used to focusthe manipulated beams 10 (a-d) as a means for obtaining experimentaldata. Using a lens (L1) 42 to focus the manipulated beams 10 (a-d) isnot a requirement for practicing a method in accordance with theprinciples of this invention. The lens was used merely for laboratorysimplicity. This focal spot was then imaged onto a camera by amicroscope objective (MO) to magnify the focal pattern.

FIG. 9 shows the results of the layout just described—a far fieldcomposite intensity distribution of the composite beam imaged during anexperiment using the apparatus generally described above to perform oneembodiment of a method in accordance with the principles of the presentinvention. There is a null at the center (46) in the far fieldsurrounded by a square grid of peaks (48 (a-d)) having an intensity thatdecreases with increasing distance from the center.

Experiments were performed according to the principles of the presentinvention using three and six manipulated beams. The layout used duringthis experiment (FIG. 10) was one embodiment of a layout constructed topractice a method in accordance with the principles of the presentinvention. Many different components could be used in accordance withthe principles of the present invention. In this particular embodiment,a continuous wave Nd:YAG laser (48) is coupled into a polarizationmaintaining single mode optical fiber (50). The signal is then directedthrough an eight channel Electro-optic Modulator (52) which createseight copies of the original signal and provides phase and amplitudecontrol on each channel. U.S. Pat. Nos. 7,187,492 and 7,058,098 provideadditional detail regarding controlling and stabilizing the opticalphase of multiple beams. Any conventional means of providing multiplebeams and controlling their phase and amplitude may be used inaccordance with the principles of the present invention. Each channel isthen directed out of an optical fiber and collimated by a 1 inch lens(L1-6) 58. The lenses 58 are spaced about 0.020″ (1.020″center-to-center) apart in a hexagonal layout. To adjust the focus andtip-tilt of each beam, a 5-axis stage (not shown) holds each fiber. Thebeams are then directed through six ½ wave plates 60 to provide theproper polarization, thereby yielding manipulated beams 62. Themanipulated beams 62 were then directed through a five (5) inchtransform lens with about a sixty inch focal distance (not shown).Finally, the focus spot is then imaged onto a camera by a microscopeobjective (not shown) in order to view the far field image.

FIGS. 3 and 5 show a composite near field intensity distribution ofthree and six manipulated beams. The simulation of a six beam layoutconfiguration is shown in FIG. 8. The far field composite intensitydistribution of a composite beam produced using six manipulated beams isshown in FIG. 11. For the axisymmetric polarization case (bottom), thereis a null at the center (64) in the far field surrounded by a hexagonalring and six more nulls (66 (a-f)) further from the center in ahexagonal grid. Notice also that there is still a reasonable correlationbetween the simulated far field patterns and the experimental far fieldpatterns.

The simulation of a three beam layout configuration is shown in FIG. 6.The far field composite intensity distribution of a composite beamproduced using three manipulated beams is shown in FIG. 12. There is anull at the center (70). Notice also that there is still a reasonablecorrelation between the simulated far field patterns and theexperimental far field patterns. However, the correlation is not asstrong because this particular experiment did not have a well stabilizedphase for each manipulated beam. Subsequent experiments will likelyinclude a feedback system—similar to the systems described in U.S. Pat.Nos. 7,187,492 and 7,058,098—to maintain the proper phase relationshipsbetween each of the beams.

Theoretical Foundations:

Consider first the propagation of only the ith Gaussian beam of an arrayusing rectangular coordinates. Assume that this beam has its waist atthe origin, travels along the z axis, and has a frequency ω. It has auniform polarization in the x-direction. Then the electric field of thiswave as a function of position has the formE _(i)({right arrow over (x)})=ε_(i)({right arrow over (x)})e^(iωt),  (1)where we define

$\begin{matrix}{{ɛ_{i}\left( \overset{\rightarrow}{x} \right)} = {A\frac{w_{0}}{w(z)}{\mathbb{e}}^{i{\lbrack{{kz} - {\tan^{- 1}{({z/z_{0}})}} + \phi^{\prime}}\rbrack}}{\mathbb{e}}^{{{{ik}{({{\Delta\; x_{i}^{2}} + {\Delta\; y_{i}^{2}}})}}/2}{R{(z)}}}{\mathbb{e}}^{{- {({{\Delta\; x_{i}^{2}} + {\Delta\; y_{i}^{2}}})}}/{w^{2}{(z)}}}}} & (2)\end{matrix}$in which Δx_(i)=x−x_(i) is the distance along the x axis from the centerof beam i at z=0 to the point x, Δy_(i)=y−y_(i) is the distance alongthey axis from the center of beam i at z=0 to the point y, A is theamplitude of the field, w₀ is the standard Gaussian beam waist, φ′ isthe phase, λ is the beam's wavelength, and z₀ is the Rayleigh rangedefined as

$\begin{matrix}{z_{0} = {\frac{\pi\; w_{0}^{2}}{\lambda}.}} & (3)\end{matrix}$R(z) and w(z) are defined as follows.

$\begin{matrix}{{R(z)} = {z + {\frac{z_{0}^{2}}{z}\mspace{14mu}{and}}}} & (4) \\{{w(z)} = {w_{0}{\sqrt{1 + \frac{z^{2}}{z_{0}^{2}}}.}}} & (5)\end{matrix}$Since this beam is a Gaussian it remains a Gaussian as it propagates.Note also from the last exponential function that the width of theGaussian extends in x and y as the beam propagates along the z axis. Theconsequence of this is that for z>>z₀ equation [2] becomes

$\begin{matrix}{{ɛ_{i}\left( \overset{\rightarrow}{x} \right)} = {A\frac{z_{0}}{z}{\mathbb{e}}^{i{({{kz} - \phi})}}{\mathbb{e}}^{{{{ik}{({x_{i}^{2} + y_{i}^{2}})}}/2}z}{\mathbb{e}}^{{{{- {z_{0}^{2}(\;{x_{i}^{2} + \; y_{i}^{2}})}}/w_{0}^{2}}z^{2}},}}} & (6)\end{matrix}$where φ=φ′−π/2.

Now if we take the case of 2 beams which are in phase (i.e. φ₁=φ₂=φ) atz=0 with x_(i) and y_(i) coordinates (1, 0) for beam 1 and (−1, 0) forbeam 2, then the total time averaged field will be

$\begin{matrix}\begin{matrix}{{ɛ_{i}\left( \overset{\rightarrow}{x} \right)} = {{ɛ_{1}\left( \overset{\rightarrow}{x} \right)} + {ɛ_{2}\left( \overset{\rightarrow}{x} \right)}}} \\{= {A\frac{z_{0}}{z}{\mathbb{e}}^{i{({{kz} - \phi})}}{\mathbb{e}}^{{{iky}^{2}/2}z}{\mathbb{e}}^{{- z_{0}^{2}}{y^{2}/w_{0}^{2}}z^{2}} \times}} \\{\left( {{{\mathbb{e}}^{{{{ik}{\lbrack{{({x - 1})}^{2} + y^{2}}\rbrack}}/2}z}{\mathbb{e}}^{{{- {z_{0}^{2}{\lbrack{{({x - 1})}^{2} + y^{2}}\rbrack}}}/w_{0}^{2}}z^{2}}} +} \right.} \\{\left. {{\mathbb{e}}^{{{{ik}{\lbrack{{({x - 1})}^{2} + y^{2}}\rbrack}}/2}z}{\mathbb{e}}^{{{- {z_{0}^{2}{\lbrack{{({x - 1})}^{2} + y^{2}}\rbrack}}}/w_{0}^{2}}z^{2}}} \right).}\end{matrix} & (7)\end{matrix}$Clearly if we look on axis we will find that the field isδ(0,0,z)=2δ₁(0,0,z)=2δ₂(0,0,z)  (8)which in the far field is clearly going to be nonzero. However if wetake the same case with φ₁=0 and φ₂=π, then the total time averagedfield will be

$\begin{matrix}\begin{matrix}{{ɛ\left( \overset{\rightarrow}{x} \right)} = {{ɛ_{1}\left( \overset{\rightarrow}{x} \right)} + {ɛ_{2}\left( \overset{\rightarrow}{x} \right)}}} \\{= {A\frac{z_{0}}{z}{\mathbb{e}}^{ikz}{\mathbb{e}}^{{{iky}^{2}/2}z}{\mathbb{e}}^{{- z_{0}^{2}}y^{{2/w_{0}^{2}}z^{2}}} \times}} \\{\left( {{{\mathbb{e}}^{{{{ik}{({x - 1})}}^{2}/2}z}{\mathbb{e}}^{{{- {z_{0}^{2}{({x - 1})}}^{2}}/w_{0}^{2}}z^{2}}} +} \right.} \\{\left. {{\mathbb{e}}^{{{{ik}{({x + 1})}}^{2}/2}z}{\mathbb{e}}^{{{- {z_{0}^{2}{({x + 1})}}^{2}}/w_{0}^{2}}z^{2}}} \right).}\end{matrix} & (9)\end{matrix}$Then along the z axis, clearlyδ(0,0,z)=0.  (10)Since the intensity is proportional to the square magnitude of theelectric field, the intensity will also be zero along the z axis. Thisis due to the fact that these two fields are out of phase and of equalmagnitude along the z axis. The resulting field distribution isanti-symmetric, while for the previous case the resulting field issymmetric with respect to the y axis. Thus for any uniform distributionwith an even number of beams, N, rotated by 360%, the intensity alongthe z axis will be zero. A generalization to odd numbers of beams andarbitrary intensity distributions follows.

To generalize the above result to an arbitrary number of beams with semiarbitrary near field intensity distributions, consider the followingdiscussion. It is sufficient to consider a Huygens approach. In freespace, light from a point source expands in a sphere and can bedescribed by the wave equation in spherical coordinates

$\begin{matrix}{{\nabla^{2}\psi} = {{\frac{1}{r^{2}}\frac{\partial\;}{\partial r}\left( {r^{2}\frac{\partial\psi}{\partial r}} \right)} = {\frac{1}{v^{2}}\frac{{\partial^{2}\psi}\;}{\partial t^{2}}}}} & (11)\end{matrix}$where r is the distance from the point source and t is the time. Theform of the solution can be shown to be

$\begin{matrix}{{{\psi\left( {r,t} \right)} = {\left( \frac{ɛ_{0}}{r} \right){\mathbb{e}}^{i{({{k \cdot r} \pm {\omega\; t}})}}}},} & (12)\end{matrix}$

where the constant δ₀ is the source strength, k is the wave vector, andω is the angular frequency. This is the basis for using the Huygensformalism which describes light as expanding spheres.

Consider then that a beam of predetermined wavelength, arbitrary shape,and arbitrary intensity distribution is placed on the perimeter ofcircle with radius d/2 in the x-y plane (z=0). Now consider an array ofN such beams (for example, see the example for N=5 in FIG. 1), where Nis an integer greater than or equal to three, evenly distributed on theperimeter of the circle 4 each oriented with the perimeter arc of thecircle 4 passing through the same corresponding portion of each beam 2(a-e). The angle 3 separating adjacent beams is θ_(N)=360°/N. The saidbeams must be parallel to each other and for this example arepropagating along the z direction. Using the Huygens expanding sphereformalism for describing one such beam of light propagating in the zdirection, an expression for the disturbance caused at a point, P, alongthe z axis can be written as

$\begin{matrix}{{{E_{n}(z)} = {{\underset{{bean}{aperture}}{\int\int}\frac{ɛ_{A}\left( {r_{n},\theta_{n}} \right)}{R_{n}}{\hat{e}}_{n}\mspace{11mu}{\mathbb{d}S_{n}}} = {\underset{{bean}{aperture}}{\int\int}\frac{ɛ_{A}\left( {r_{n},\theta_{n}} \right)}{\sqrt{r_{n}^{2} + z^{2}}}{\hat{e}}_{n}\mspace{11mu}{\mathbb{d}S_{n}}}}},} & (13)\end{matrix}$which is proportional to the electric field and in which we have chosena cylindrical coordinate system. Here δ_(A)(r_(n),θ_(n)) is the strengthper unit area (analogous to δ₀ mentioned above) of the light from thenth beam 2 (a-e), θ_(n)=η⁰−360°·n/N is the angle from the x axis aroundthe circle to the nth beam 2 (a-e), ê_(n) 17 (a-e) is the polarizationunit vector of the nth beam 2 (a-e), the beam aperture is the crosssection of the nth beam 2 (a-e) in the z=0 plane, and dS_(n) 5 (a-e) isan infinitesimal area for integration over the nth beam aperture. Forsimplicity δ_(A)(r_(n),η_(n)) is taken to be a well behaved function. Tofind the complete on-axis disturbance from such an array at a distance zalong the axis, one must calculate the sum of the individualdisturbances

$\begin{matrix}{{E(z)} = {{\sum\limits_{n = 1}^{N}\;{E_{n}(z)}} = {\sum\limits_{n = 1}^{N}{\underset{{bean}{aperture}}{\int\int}\frac{ɛ_{A}\left( {r_{n},\theta_{n}} \right)}{\sqrt{r_{n}^{2} + z^{2}}}{\hat{e}}_{n}\mspace{11mu}{{\mathbb{d}S_{n}}.}}}}} & (14)\end{matrix}$In general for x and y not on the z axis, this integration and summationwould be complex to calculate, but because of the on-axis symmetry ofthe beam arrangement, simplifications can be made by choosing dS_(n) 5(a-e) to be in the same relative location in each aperture. Then R₁=R₂=. . . =R_(n)=R 11 (a and b) and by extension r₁=r₂= . . . =r_(n)=r 7 (aand b). It also can be seen that the integration over each aperture isidentical, and thus the previous expression can be rewritten as

$\begin{matrix}{{E(z)} = {{N \cdot \underset{{bean}{aperture}}{\int\int}}\frac{ɛ_{A}\left( {r_{n},\theta_{n}} \right)}{\sqrt{r_{n}^{2} + z^{2}}}\left( {\sum\limits_{n = i}^{N}{\hat{e}}_{n}}\mspace{11mu} \right){{\mathbb{d}S}.}}} & (15)\end{matrix}$where the expression in equation [14] has been multiplied by N and thesummation is only over the unit vectors 17 (a-e) of the beams. Thesummation of the polarization unit vectors 17 (a-e) in this expressioncan be shown to cancel which causes E(z)=0 at any point along thepositive z axis (i.e. the intensity on axis is zero creating a centralnull in the composite beam FIG. 6, 26; FIG. 7, 28; FIG. 8, 30; FIG. 9,46; FIG. 11, 64; FIG. 12, 70). To more carefully consider thiscancellation, refer to FIG. 13. The unit vectors of each beam are shownfor the cases N=4 (74 a), N=5 (74 b), and N=6 (74 c). By rearranging theunit vectors head-to-tail for the cases N=4 (76 a), N=5 (76 b), and N=6(76 c), it can be clearly seen that the polarization vectors cancel forN=4, 5, and 6. Since η=η₀−360°/N is the formula both for thepolarization unit vector of each succeeding beam rotated by η₀ as wellas for the external angles of a regular polygon rotated by η₀, it isclear that this argument can be extended to any number of beams, N. Thusfor the geometry described above for an arbitrary number of beams,E(z)=0 for any point along the z axis for z>0.

The disturbance created by the beams in the near field also goes to zerofar from the z axis in a plane of constant z (FIG. 1, 19) containingpoint P (FIG. 1, 9). If we define w_(g) (η,z) to be the generalizedextent of an individual beam at a point (r,η,z) such that the greatmajority of the beam falls within the boundary it defines, then clearlyat any distance z along the axis, the disturbance E(r, z) goes to 0 forr>>w_(g)(η,z).

The disturbance can also be shown to maintain its cylindrical symmetryin the plane where z is constant (FIG. 1, 19) containing point P (9).Consider a circle (21) of radius ρ (23) in the said plane (19).Temporarily ignoring the oscillatory nature of light, it can clearly beseen that contributions to the total disturbance from the dS_(n)'s 5(a-e) at any point on the perimeter circle are strongest from thedS_(n)'s 5 (a-e) which are closer to the point on the perimeter (21)under consideration. Thus as one progresses around the perimeter of thecircle (21) in the said plane (19), the cylindrical symmetry from theapertures at z=0 is imposed on the composite beam cross section at thesaid plane (19). Another way to express this observation is to say thatin the absence of anything to break the symmetry, the cylindricalsymmetry of the near field is carried on in the composite beam as itpropagates.

A cylindrical vector or CV beam is a beam that has cylindrical symmetry.The discrete nature of the manipulated beams in the near field describedin this Theoretical Background section will create oscillatorydiffraction effects on the circle (FIG. 1, 21), but nevertheless thediscrete cylindrical symmetry of the near field array yields adiffraction, pattern that is zero on axis as well as far from the axisand maintains cylindrical symmetry—that is, it approximates aninfinitely symmetric CV beam by using N discrete beams. These discretecylindrical vector beams or DCV beams will approximate a CV beam withinfinite-fold or continuous cylindrical symmetry. The quality of thisapproximation will depend on factors such as the number of elements, thenear field fill factor of the beams, and the quality of the phasing ofthe beams.

The foregoing discussion did not limit the distance of point, P, fromthe array of beams at z=0. However in many practical situations, it isconvenient to consider the far field images formed from beams which haverelatively uniform phase across each aperture in the near field,commonly referred to as the beams being collimated. Point P is said tobe in the far field when the distance between the array of beams in thenear field at z=0 and P is greater than a²/λ where a is the aperturesize of the array under consideration—in this case it is comparable to d(two times the radius d/2 in FIG. 1)—and λ is the wavelength of thepropagating light. The shape of composite beams in the far field can becreated at a much shorter distance by using a lens commonly referred toas a transform lens. By using a transform lens, far field measurementscan be made at the focal plane of the lens. The near field refers to anypoint between the plane containing P and the plane defined by z=0 whenthe said point is close to the z=0 plane compared with the far fielddistance. It is convenient to consider composite beams in the far fieldbecause, other than continuing to expand with increasing z, thecomposite intensity distributions do not continue to evolve as zincreases, while the composite distribution changes significantlybetween z=0 and the far field. With this description consider thefollowing discussion of calculations and measurements done to verify theconcepts herein described.

FIGS. 14-19 illustrate the composite intensity distributions ofcomposite beams which have a uniform linear polarization (unlike acomposite beam produced in accordance with the principles of the presentinvention) lying along the same direction for each beam as in theexample leading up to equation 8. Consider the case of three, four andsix beams arranged with a near field composite intensity distributionsimilar to the one depicted in FIG. 3. In this example the wavelength λof the light is about 1.06 nm, the distance d is about 7.62 cm, and thesetup used for measurements is similar to the one shown in FIG. 10 withthe far field conditions being created using a transform lens. For thecase of three Gaussian beams polarized in the same direction(left-to-right), the calculated far field composite intensitydistribution is depicted in FIG. 14 while the corresponding measuredcomposite intensity distribution is depicted in FIG. 15. The centers ofthe composite intensity distributions depicted—both calculated andmeasured—are clearly maximum rather than null. However the DCV case witheach beam having a uniform linear polarization along the direction fromthe center of the distribution outward (refer to FIG. 3), the calculatedfar field composite intensity distribution is depicted in FIG. 6 whilethe corresponding measured intensity distribution is depicted in FIG.12. In this case, the centers of the far field composite distributionsare clearly null rather than maximum. It should also be noted that inthat particular embodiment, the standard far field composite intensitydistributions are characterized by peaks in a hexagonal grid (FIGS. 14and 15) while the DCV far field composite intensity distributions arecharacterized by nulls in a hexagonal grid (FIGS. 6 and 12). The casesfor four and six beams were also tested and are shown in FIGS. 16-19.The case for six beams was identical to the case described above forthree beams, while the case for four beams differed by having a distanced of about 2.41 cm and a setup similar to the one shown in FIG. 2. Thecase of four beams in the near field (FIG. 4) yields calculated andmeasured standard far field composite intensity distributions shown inFIGS. 16 and 17 respectively with calculated and measured DCV far fieldcomposite intensity distributions shown in FIGS. 7 and 9 respectively.The case of six beams in the near field (FIG. 5) yields calculated andmeasured standard far field composite intensity distributions shown inFIGS. 18 and 19 respectively with calculated and measured DCV far fieldcomposite intensity distributions shown in FIGS. 8 and 11 respectively.Again as with three beams, it should also be noted that the standard farfield composite intensity distributions for four and six beams arecharacterized by peaks in the respective grid (FIGS. 16, 17, 18, and 19)while the DCV far field composite intensity distributions arecharacterized by nulls in the respective grid (FIGS. 7, 9, 8, and 11).

In the experimental and simulated cases, each beam was cylindricallysymmetric. As such, the beams in the near field were not rotated byη=360°/N since the cylindrical symmetry of each beam would be nodifferent under rotations. Hence, it was sufficient to simply rotate thepolarization of each beam. However an asymmetrical beam or a beam withlower symmetry might be required to be physically rotated as shown inFIG. 1 in order to satisfy the conditions to be a DCV beam. Some specialcases clearly exist, such as using triangular beams where N=3, squarebeams where N=4, and hexagonal beams where N=6, etc. when the beamsrequire only the polarization to be rotated providing the beams all havethe same orientation and are evenly distributed around the perimeter ofa circle at z=0. Again these simplifications arise from the inherentsymmetry in each case while beams with no inherent symmetry wouldrequire rotation following the example in FIG. 1.

In addition to having a single DCV beam arising from a single near fieldarray, it is possible to nest multiple concentric near field arrays togenerate a more complex DCV beam. Namely, the existence of more than oneconcentric near field array each following the principles of the presentinvention still creates a DCV beam which is a mixture of the individualDCV beams arising from the individual concentric near field DCV arrays.This can be clearly seen by considering the following. Each DCV beamindividually has a central null. If these central nulls are placedconcentrically around a common axis, there will still exist a centralnull. Similarly, the intensity of each DCV beam will also go to zero atsome distance from the axis which means that some w_(g) (η,z) can stillbe created which will describe the size of the beam. The cylindricalsymmetry of the total DCV beam will also be preserved since individuallythey are each concentric and cylindrically symmetric.

1. A process for producing annular far-field patterned beams,comprising: providing a plurality of optical beams; manipulating saidplurality of optical beams; producing an odd-numbered plurality ofmanipulated beams from said plurality of optical beams; providing eachof said odd-numbered plurality of manipulated beams a selected nearfield shape such that each of said odd-numbered plurality of manipulatedbeams has a near field geometry, wherein said near field geometry ofeach of said odd-numbered plurality of manipulated beams is the same;providing each of said odd-numbered plurality of manipulated beams apolarization that differs from the polarization of its adjacent of saidodd-numbered plurality of manipulated beams by an amount equal to 360degrees divided by a total number of said odd-numbered plurality ofmanipulated beams; providing each of said odd-numbered plurality ofmanipulated beams with a selected wavelength such that each of saidodd-numbered plurality of manipulated beams has the same wavelength;providing each of said odd-numbered plurality of manipulated beams witha selected phase such that each of said odd-numbered plurality ofmanipulated beams is in phase with each other said odd-numberedplurality of manipulated beams; providing a distribution of saidodd-numbered plurality of manipulated beams such that each of saidodd-numbered plurality of manipulated beams is uniformly distributed onthe perimeter of a single circle and the composite near field intensitydistribution of said odd-numbered plurality of manipulated beams has acentral null; orienting each of said uniformly distributed odd-numberedplurality of manipulated beams on the perimeter of said single circlesuch that the single circle passes through an identical portion of eachof said odd-numbered plurality of manipulated beams' intensitydistribution; propagating said odd-numbered plurality of manipulatedbeams having said near field distribution, said near field geometry,said wavelength, said polarization, said phase, said distribution andsaid orientation such that each of said odd-numbered plurality ofmanipulated beams is propagated in parallel with each other of saidodd-numbered plurality of manipulated beams; and producing a compositebeam comprising said odd-numbered plurality of manipulated beams,wherein said composite beam has a composite far field intensitydistribution having a central null and discrete cylindrical symmetry. 2.The process of claim 1 wherein said odd-numbered plurality ofmanipulated beams comprises at least five manipulated beams.
 3. Theprocess of claim 1 further comprising the step of passing a single beamthrough a diffractive optical element to produce said plurality ofoptical beams wherein said plurality of optical beams comprises an arrayof beams.
 4. The process of claim 1 wherein said near field geometry ofsaid odd-numbered plurality of manipulated beams is a cylindricallysymmetrical geometry.
 5. The process of claim 1, wherein saidodd-numbered plurality of manipulated beams is three odd-numberedplurality of manipulated beams.
 6. The process of claim 1, wherein eachof said plurality of optical beams is single mode Gaussian.
 7. Theprocess of claim 1, wherein each of said odd-numbered plurality ofmanipulated beams is single mode Gaussian.
 8. The process of claim 6,wherein each of said odd-numbered plurality of manipulated beams issingle mode Gaussian.
 9. The process of claim 2, wherein each of saidplurality of optical beams is single mode Gaussian.
 10. The process ofclaim 2, wherein each of said odd-numbered plurality of manipulatedbeams is single mode Gaussian.
 11. The process of claim 9, wherein eachof said odd-numbered plurality of manipulated beams is single modeGaussian.
 12. The process of claim 3, wherein each of said plurality ofoptical beams is single mode Gaussian.
 13. The process of claim 3,wherein each of said odd-numbered plurality of manipulated beams issingle mode Gaussian.
 14. The process of claim 12, wherein each of saidodd-numbered plurality of manipulated beams is single mode Gaussian. 15.The process of claim 4, wherein each of said plurality of optical beamsis single mode Gaussian.
 16. The process of claim 4, wherein each ofsaid odd-numbered plurality of manipulated beams is single modeGaussian.
 17. The process of claim 15, wherein each of said odd-numberedplurality of manipulated beams is single mode Gaussian.
 18. The processof claim 5, wherein each of said plurality of optical beams is singlemode Gaussian.
 19. The process of claim 5, wherein each of saidodd-numbered plurality of manipulated beams is single mode Gaussian. 20.The process of claim 18, wherein each of said odd-numbered plurality ofmanipulated beams is single mode Gaussian.